1. RPI / Geo* / NGA & DARPA Oct 1-2 2007 1 DEM Compression and Terrain Approximation; Smugglers and Border Guards Prof W Randolph Franklin, Prof Frank Luk, Prof Barbara Cutler, Prof Marcus Andrade, Metin Inanc, Zhongyi Xie, Dan Tracy, Jon Muckell Rensselaer Polytechnic Institute mail@wrfranklin.org, 703-447-7808

2. RPI / Geo* / NGA & DARPA Oct 1-2 2007 2 Complete Record This version contains both the slides that I showed to NGA (on 10/1/2007), and to DARPA (on 10/2/2007). Therefore it is both a good introductory and detailed record of RPI’s performance to date. http://www.ecse.rpi.edu/~wrf/pmwiki/GeoStar contains most material ever given by RPI to DARPA or NGA. Ask WRF for a password

3. RPI / Geo* / NGA & DARPA Oct 1-2 2007 3 Team Prof Randolph Franklin – helping everyone Prof Barbara Cutler – computer graphics Prof Frank Luk (on leave as Vice-President (Academic) of Hong Kong Baptist U) – numerical analysis Prof Marcus Andrade – visiting from UF Vi Ç osa (Brazil) – computational geometry Metin Inanc – ODETLAP Zhongyi Xie – ODETLAP Dan Tracy – multiobserver siting, path planning Jon Muckell – hydrology. Changes

4. RPI / Geo* / NGA & DARPA Oct 1-2 2007 4 Potential Benefits More compact terrain representations. Store the ever larger amounts of terrain data. Spend time on the compression (which is done once) to get most compact representation. Works on 16 bit topography. Conflate overlapping inconsistent cells. Overlay large, low precision cell by smaller high precision cells => one unified elevation field. Efficient routing by nonspecialists. Route our aircraft away from antiaircraft batteries Better hydrographic computations. Flood prediction.

5. RPI / Geo* / NGA & DARPA Oct 1-2 2007 5 An Inadequate Terrain Representation

6. RPI / Geo* / NGA & DARPA Oct 1-2 2007 6

7. 7 Goal 1: DEM Compression and Terrain Approximation ODETLAP: alternate terrain representation. Compact. Allows lossy - size / quality tradeoffs. Emphasized decompression speed. Evaluated on visibility, mobility metrics.

8. RPI / Geo* / NGA & DARPA Oct 1-2 2007 8 Milestone Progress Phase I: 10x compression while maintaining usefulness; Phase II: 100x Reverse engineered HRTI Analysis Tool’s slope formula to avoid running HAT each time. We gave before-and-after data to NGA demonstrating this. Further improvements made since then. Fitting other components of proposal (e.g., hydrology) to correspond to milestones.

9. RPI / Geo* / NGA & DARPA Oct 1-2 2007 9 Key Differentiating Factors Smooth representation no visible blockiness Allows progressive transmission longer transmission => more accurate reconstruction Conflates inconsistent partially overlapping data. Interpolates partial sets of elevation posts generates continuous slopes even when the input data consists of nested contour lines. Infers local maxima inside the topmost contour.

10. RPI / Geo* / NGA & DARPA Oct 1-2 2007 10 ODETLAP Process Since our first description of ODETLAP at the 1998 Spatial Data Handling Symposium, we’ve built this system.

11. RPI / Geo* / NGA & DARPA Oct 1-2 2007 11 ODETLAP Point Selection Several options: Incremental TIN to find most important points, then greedy insertion of worst points (Allows progressive transmission) Regular grid of points (more points, but compress better) (More compact) NEW Stream and ridgeline points (Preliminary) NEW

12. RPI / Geo* / NGA & DARPA Oct 1-2 2007 12 Other Point Selection Strategies The following were not as good: Select highly visible points, or Random points, or Points based on histogram of heights with boosted sampling for less frequent elevation bands and small connected components.

13. RPI / Geo* / NGA & DARPA Oct 1-2 2007 13 Incremental Triangulated Irregular Network (TIN) Can process 10 8 points on a laptop. Works in memory w/o needing to page data from disk. Inserts points incrementally, in order of importance. Can progressively transmit terrain. Identifies ridge lines automatically.

14. RPI / Geo* / NGA & DARPA Oct 1-2 2007 14 Coding the Points to Reduce Space Code (x,y) separately from z. NEW If (x,y) a regular grid: give its resolution Else: run-length encode the bitmap. 0100000011000010001 -> 16043 Only about 1000 of 160,000 bits are 1. Z: delta code, then bzip2. 100 125 90 90 100 -> 100 25 -35 0 10

15. RPI / Geo* / NGA & DARPA Oct 1-2 2007 15 Traveling Salesman Path NEW! Hypothesis: nearby points often have nearby Z, which delta code better Find a traveling salesman path through the selected points. Put the Z in that order and code them. (X,Y) coding is not affected. Status: have some preliminary results.

16. RPI / Geo* / NGA & DARPA Oct 1-2 2007 16 Info theoretic min for (x,y) Assume that 1000 of 400x400 bits are 1, rest are 0. Assuming no structure in the 1s, size is lg(choose(160000,1000)) = 8754 bits We approach that within 20%. That’s why we separate (x,y) from (z).

17. RPI / Geo* / NGA & DARPA Oct 1-2 2007 17 Better Than the Info-Theor Limit? The information theoretic limit was calculated assuming no structure in the points. Is there a structure to exploit? Scooping Grids of points

18. RPI / Geo* / NGA & DARPA Oct 1-2 2007 18 Reconstruction Context Extension of classical Laplacian partial differential equation used to solve heat flow etc Now possible with new numerical computation techniques on large sparse overdetermined systems of linear equations Adds capabilities to the classical system Local maxima inference Inconsistent data conflation

19. RPI / Geo* / NGA & DARPA Oct 1-2 2007 19 ODETLAP Point Reconstruction Solve an overdetermined variant of a Laplacian PDE. Known pts: z ij = h ij All pts: 4z ij = z i-1j + z i+1j + z ij-1 + z ij+1 Easily processes 400x400 arrays of elevation posts in Matlab (160,000 unknowns) Process larger arrays with Page-Saunders technique

20. RPI / Geo* / NGA & DARPA Oct 1-2 2007 20 ODETLAP on Larger Cells We could go to Page-Saunders if there is interest. My masters student John Childs did this in 2003, before Geo*. Goal: several-thousand-square cell.

21. RPI / Geo* / NGA & DARPA Oct 1-2 2007 21 Four Matlab Interpolation Techniques on Nested Square Contours This difficult example was chosen to illustrate all these methods’ limitations. Inverse distanceDelauney, linear Delauney, cubicDelauney, nearest neighbor

22. RPI / Geo* / NGA & DARPA Oct 1-2 2007 22 ODETLAP on Nested Squares Surface now looks much better. Can tradeoff accuracy vs smoothness. R=0.1, mean err=0.01% R=1.0, mean err=0.8%R=3.0, mean err=2.7% Original data

23. RPI / Geo* / NGA & DARPA Oct 1-2 2007 23 Terrain Test Data Extracted from level 2 DEMs Elevation range

24. RPI / Geo* / NGA & DARPA Oct 1-2 2007 24 Hill1

25. RPI / Geo* / NGA & DARPA Oct 1-2 2007 25 Hill2

26. RPI / Geo* / NGA & DARPA Oct 1-2 2007 26 Hill3

27. RPI / Geo* / NGA & DARPA Oct 1-2 2007 27 Mtn1

28. RPI / Geo* / NGA & DARPA Oct 1-2 2007 28 Mtn2

29. RPI / Geo* / NGA & DARPA Oct 1-2 2007 29 Mtn3

30. RPI / Geo* / NGA & DARPA Oct 1-2 2007 30 Accuracy Metrics Since flatter cells are easier, Following slides and tables show RMS error, meters, or (RMS error) / (elevation range in the cell) Slope computed using Zevenbergen- Thorne algorithm used in NGA HAT. Slope error is always RMS degrees.

31. RPI / Geo* / NGA & DARPA Oct 1-2 2007 31 TIN + Greedy ODETLAP Results Data Size, bytes Com- pression ratio RMS Elev Error, m RMS Slope Error, deg hill11880170:12.833.53 hill21962163:14.068.06 hill31739184:11.661.65 mtn11979162:13.7714.0 mtn22006160:14.3114.1 mtn32004160:14.5813.3

32. RPI / Geo* / NGA & DARPA Oct 1-2 2007 32 TIN+Greedy Elevation Accuracy

33. RPI / Geo* / NGA & DARPA Oct 1-2 2007 33 TIN+Greedy Slope Accuracy

34. RPI / Geo* / NGA & DARPA Oct 1-2 2007 34 TIN+Greedy Elevation Comparison Mtn2 Dataset 7641 bytes => 42:1 compression ratio

35. RPI / Geo* / NGA & DARPA Oct 1-2 2007 35 Zevenbergen-Thorne Slope Comparison Mtn2 Dataset 7641 bytes => 42:1 compression ratio

36. RPI / Geo* / NGA & DARPA Oct 1-2 2007 36 Even Better Slope Representation Idea: Extend the ODETLAP equations directly to incorporate the original representation’s vector gradient at critical points, instead of inferring slope from adjacent elevations. Status: being designed.

37. RPI / Geo* / NGA & DARPA Oct 1-2 2007 37 Different Point Selection Strategies Previous slides used TIN+greedy That can be made to allow progressive transmission of the points, by replacing bitmap coding of the (X,Y) with a bzip2 compression. Following slides use regular grid point selection. That compresses better but doesn’t do progressive transmission.

38. RPI / Geo* / NGA & DARPA Oct 1-2 2007 38 Regular Grid ODETLAP Results Dataset# Points Com- pressed Size Com- pression Ratio Elev RMS (m) Slope RMS (deg) Hill15293061046:19.634.32 Hill21600807397:19.986.54 Hill32251721860:19.713.04 Mtn144892194146:19.6610.13 Mtn244892027158:19.9510.34 Mtn344892013159:19.919.85 (Our second ODETLAP point insertion strategy)

39. RPI / Geo* / NGA & DARPA Oct 1-2 2007 39 Regular Grid ODETLAP Accuracy

40. RPI / Geo* / NGA & DARPA Oct 1-2 2007 40 Merged Point Selection Strategy Start with regular grid Greedily add more points, Use overlapping local grids, like with differential geometry coordinate frames. Status: being designed.

41. RPI / Geo* / NGA & DARPA Oct 1-2 2007 41 Missing Data Fillin ODETLAP can fill in missing circles with r<=100. Slopes are continuous across the boundary. Contours are realistic. Next slide compares ODETLAP to 3 Matlab methods.

42. RPI / Geo* / NGA & DARPA Oct 1-2 2007 42 Fillin Comparison ODETLAPLaplacianThin plate Matlab nearestMatlab linearMatlab cubic

43. RPI / Geo* / NGA & DARPA Oct 1-2 2007 43 Packaging ODETLAP Current process is a group of programs combining Matlab, bzip2, C++. Being packaged into a unified system for distribution to NGA. We can complete this when needed.

44. RPI / Geo* / NGA & DARPA Oct 1-2 2007 44 More Terrain Representations Scooping still has the greatest longterm potential. Could use RPI’s BlueGene/L, the 2 nd fastest computer in a university setting in the world. (source: http://top500.org/lists/2007/06)http://top500.org/lists/2007/06 Problem: It doesn’t run Matlab.

45. RPI / Geo* / NGA & DARPA Oct 1-2 2007 45 Goal 2: Smugglers and Border Guards (aka Siting & Path Planning)

46. RPI / Geo* / NGA & DARPA Oct 1-2 2007 46 Multiobserver siting steps 1. Compute approximate visibility index of every possible observer. 2. Compute exact viewsheds of the best. 3. Greedily insert potential observers into the final set of observers, maintaining a bitmap of the cumulative viewshed. 4. Intervisibility => insert only visible observers. Key: fast bitmap operations allow hundreds of observers to be sited with hi-res viewsheds.

47. RPI / Geo* / NGA & DARPA Oct 1-2 2007 47 Sample Viewsheds Note the level of detail

48. RPI / Geo* / NGA & DARPA Oct 1-2 2007 48 Viewshed uncertainty

49. RPI / Geo* / NGA & DARPA Oct 1-2 2007 49 With or w/o intervisibility Color -> elevation; Black -> hidden. Intervisibility enforced No intervisibility required

50. RPI / Geo* / NGA & DARPA Oct 1-2 2007 50 Siting Toolkit by ESRI ArcGIS DLL toolkit: an operational class configurable to perform siting simulations on any platform with a C++ compiler. Includes an ArcMap command application on Windows, to demonstrate its capabilities. Both are functional and scalable. Marquee Tool:

51. RPI / Geo* / NGA & DARPA Oct 1-2 2007 51 ArcGIS DLL Dialog Box

52. RPI / Geo* / NGA & DARPA Oct 1-2 2007 52 Path Planning (Smugglers) Find cheapest path between source and goal. Cost metric is not simply path length: (Distance) (Climbing costs) (BIG penalty for being seen)

53. RPI / Geo* / NGA & DARPA Oct 1-2 2007 53 Path planning algorithm Designed for hi-res, say 1000x1000, maps. Impossible to form the 10 6 x10 6 cost matrix. Use A* to search for initial feasible, good, path. Iterate to optimize it. Doesn’t hang up on local optima. Compute many paths to evaluate compression throughout the terrain. Note how complex our paths are. Video: multipath.wmv

54. RPI / Geo* / NGA & DARPA Oct 1-2 2007 54 Many Paths on Each Dataset

55. RPI / Geo* / NGA & DARPA Oct 1-2 2007 55 Many Paths on Hill1 Computed between 50 pairs of random start/end points

56. RPI / Geo* / NGA & DARPA Oct 1-2 2007 56 Many Paths on Mtn1

57. RPI / Geo* / NGA & DARPA Oct 1-2 2007 57 Many Paths on Mtn3

58. RPI / Geo* / NGA & DARPA Oct 1-2 2007 58 Path traversal video RPI-path-planning.wmv

59. RPI / Geo* / NGA & DARPA Oct 1-2 2007 59 Alternate Terrain Representation Evaluation Using Path Planning Q: Our alternate compressed representation has a good RMS elevation error. However, is it good for more sophisticated operations, like path planning? If we compute a smugglers path on terrain stored in our alternate rep, how good is it really? It’s not important if a computed path is very different from the optimal path, provided that its true cost is not much more expensive than the optimal path.

60. RPI / Geo* / NGA & DARPA Oct 1-2 2007 60 Smugglers Path Evaluation of ODETLAP Size: size of compressed dataset in bytes. Original binary size=320KB. Incr. cost: extra cost of optimal path computed on compressed dataset and evaluated on original dataset compared to optimal path computed on original dataset. DataSize Compr. Ratio Incr. Cost hill11763182:15.5% hill21819176:16.1% hill31607199:14.4% mtn11925166:119.2% mtn21884170:118.2% mtn31946164:117.0%

61. RPI / Geo* / NGA & DARPA Oct 1-2 2007 61 Path Planning for Road Construction Goal: Construct an optimal road connecting two points. Allowed: Material removal and deposition. Constraint: Max allowable slope is bounded. Objective function: Amount of material moved. Method: A*

62. RPI / Geo* / NGA & DARPA Oct 1-2 2007 62 Several Computed Roads

63. RPI / Geo* / NGA & DARPA Oct 1-2 2007 63 Key Differentiating Factors – Smugglers and Border Guards Can site hundreds of observers on the terrain. Observers' radius of interest may be several hundred pixels. Visibility and navigation tool optimizes a sophisticated objective function consisting of the path's total distance, the amount of uphill travel, and the distance spent in sight of any observer. Paths may be thousands of pixels long. Validates ODETLAP: Good RMS elevation error but wait, there’s more! Can be used accurately to site observers and plan paths.

64. RPI / Geo* / NGA & DARPA Oct 1-2 2007 64 Hydrology Problem Compute streams from terrain, … Assuming water flows from each cell to lower neighbors. Problem: many cells are local minima trapping flow Why? Data errors; insufficient sampling Effect: No long streams. Just starting this Phase II task in the proposal.

65. RPI / Geo* / NGA & DARPA Oct 1-2 2007 65 Common Solution Simulate gradually filling in the local minima until the water flows over the edge. But: This is slow Real-world implementation of this technique with the Taum Sauk reservoir; before and after.

66. RPI / Geo* / NGA & DARPA Oct 1-2 2007 66 Our Techniques Identify sinks & watershed boundaries with our very fast connected components program. Solve for water flow as a sparse system of linear equations. Invert elevations and solve for ridge rivers. Merge watersheds not blocked by significant ridges. Input ridge and stream points to ODETLAP.

67. RPI / Geo* / NGA & DARPA Oct 1-2 2007 67 Fast Connected Components Original application: Small block of concrete is stressed to failure while being CAT-scanned in Brookhaven synchrotron. First breaks into a few large blocks, … Need the 3D structure to understand failure. Compute the connected components of the threshholded 1000x1000x1000 CAT scan. Next slide: slices in 3 different directions look quite different.

68. RPI / Geo* / NGA & DARPA Oct 1-2 2007 68 Slices in 3 Directions of Cracking Concrete Block

69. RPI / Geo* / NGA & DARPA Oct 1-2 2007 69 Method Good implementation of union-find. Fundamental data structure is a 1-D run of solid voxels. (assumes data coherence) Carefully designed, small and fast. Form connected components with a few passes through the data.

70. RPI / Geo* / NGA & DARPA Oct 1-2 2007 70 Largest Test Run Input: 1024x1088x1088=1,212,153,856 voxels, 50% empty. 20,216,828 1-D runs, averaging 30 voxels. Output: 4,539,562 components, averaging 4.5 runs. Largest component: 2993 runs, volume 23675. Many components: only one run and two voxels. Implementation: 2GHz IBM t43p laptop, linux, gcc. Virtual memory used: only 340MB Elapsed time: 51 CPU seconds. Times scale down:  0.1 secs for 100x100x100. Largest 2D test: 19000x19000

71. RPI / Geo* / NGA & DARPA Oct 1-2 2007 71 Connected Components to Detect Sinks Identified Sinks Drainage Network

72. RPI / Geo* / NGA & DARPA Oct 1-2 2007 72 Watershed Boundary Drainage Network

73. RPI / Geo* / NGA & DARPA Oct 1-2 2007 73 Merging Watersheds not Blocked by Significant Ridges

74. RPI / Geo* / NGA & DARPA Oct 1-2 2007 74 => Larger more realistic watersheds and drainage networks Before mergingAfter merging

75. RPI / Geo* / NGA & DARPA Oct 1-2 2007 75 Summary Represent terrain in 1% of original binary space with compression ratios of 80:1 to 500:1 with 10m elevation and 5-10 degree slope error. Site multiple observers (“border guards”) and then plot smugglers paths to avoid them. Compute on the compressed terrain. Modify terrain to improve hydrology.